The Quasi-periodic Solution For The Fractional Nonlinear Schr(?)dinger Equation

In this paper,we consider the existence of quasi-periodic solutions for the fractional nonlinear Schr（?）dinger equationiψ_t=（-Δ）^sψ+f（|ψ|²ï¼‰Ïˆ,x∈T=R/Z（0.2）subject to Dirichlet boundary conditionsψ（t,0）=ψ（t,1）=0 andψ（t,-x）=-ψ（t,x）,where（-Δ）^sis the spectrally defined fraction Laplacian and s∈S=（₂¹,1）,i.e.the s-th power of the Laplace-Beltrami operator-Δ,f is a analytic function in a neighbor-hood of 0.The main conclusion of this paper is:with s as the parameter,for the small initial value,denote the set S which element are not satisfy the nonresonance condition,the measure of S is close to zero,for any s∈S\S,there exists the quasi-periodic solutions for equation（0.2）.In this paper,the KAM theory is used to prove this conclusion,The concrete steps are as follows:firstly,introducing a transformation that transform equation（0.2）into an infinite dimensional Hamiltonian system and prove that the solution of the original equation is equivalent to that of the new Hamiltonian system.Secondly,applying the theory of Birkhoff normal form to the infinite dimensional Hamiltonian system,using the symplectic coordinate transformation generated by the flow of Hamiltonian vector-field,the Hamiltonian function corresponding to the Hamiltonian system is transformed into a fourth-order Birkhoff normal form.In the infinite dimensional KAM theory,we take theν^th-step of KAM iterative as an example.We need to solve the homological equation.In this process,the frequency of the Hamiltonian system needs to satisfy the nonresonance condition.Then do KAM iteration and give the specific iterative coefficient relationship.We also prove that the composition of iterations is convergent and it is similar to the identical transformation.Finally,it is proved that the measure of the set S which is composed of the parameters s that do not satisfy the nonresonance condition is small.